For what values of $\;$ $a$ $\;$ do the equations $x^2 + ax + 1 = 0$ and $x^2 + x + a = 0$ have a root in common?
Given quadratic equations: $\;$ $x^2 + ax + 1 = 0$ $\;\;\; \cdots \; (1)$
$x^2 + x + a = 0$ $\;\;\; \cdots \; (2)$
Let $\alpha$ be the common root of equations $(1)$ and $(2)$.
Then $\alpha$ satisfies both equations $(1)$ and $(2)$.
$\therefore \;$ We have,
$\alpha^2 + a \alpha + 1 = 0$ $\;\;\; \cdots \; (3)$; $\;\;$ $\alpha^2 + \alpha + a = 0$ $\;\;\; \cdots \; (4)$
$\implies$ $\alpha^2 + a \alpha + 1 = \alpha^2 + \alpha + a$
i.e. $\;$ $a \alpha + 1 = \alpha + a$ $\;\;$ provided $\;$ $\alpha \neq 0$
i.e. $\;$ $\alpha \left(a - 1\right) = a - 1$
i.e. $\;$ $\alpha = 1$ $\;$ if $\;$ $a - 1 = 0$ $\;$ i.e. $\;$ $a \neq 1$
Substituting the value of $\alpha$ in equation $(3)$ gives
$1 + a + 1 = 0$
i.e. $\;$ $a = -2$